Abstract

This paper considers the pricing of a European
option using a
(𝐵,𝑆)-market
in which the stock price and the asset in the riskless bank
account both have hereditary price structures described by
the authors of this paper (1999). Under the smoothness assumption of the
payoff function, it is shown that the infinite dimensional
Black-Scholes equation possesses a unique classical solution. A
spectral approximation scheme is developed using the Fourier
series expansion in the space 𝐶[−ℎ,0] for the Black-Scholes equation. It is also shown that the 𝑛th approximant resembles the classical Black-Scholes equation in finite
dimensions.

1. Introduction

The pricing of contingent claims in the continuous-time financial market that consists of a bank account and a stock account has been a subject of extensive research for the last decades. In the literature (e.g., [1–5]), the equations that describe the bank account and the price of the stock are typically written, respectively, as𝑑𝐵(𝑡)=𝑟𝐵(𝑡)𝑑𝑡,𝐵(0)=𝑥,𝑑𝑆(𝑡)=𝛼𝑆(𝑡)𝑑𝑡+𝜎𝑆(𝑡)𝑑𝑊(𝑡),𝑆(0)=𝑦,(1.1)
where 𝑊={𝑊(𝑡),𝑡≥0} is a one-dimensional standard Brownian motion defined on a complete filtered probability space (Ω,𝐹,𝐏;{𝐹(𝑡),𝑡≥0}) and 𝑟, 𝛼, and 𝜎 are positive constants that represent, respectively, the interest rate of the bank account, the stock appreciation rate, and the stock volatility rate. The financial market that consists of one bank account and one stock account will be referred to as a (𝐵,𝑆)-market, where 𝐵 stands for the bank account and 𝑆 stands for the stock.

A European option contract is a contract giving the buyer of the contract the right to buy (sell) a share of a particular stock at a predetermined price at a predetermined time in the future. The European option problem is, briefly, to determine the fee (called the rational price) that the writer of the contract should receive from the buyer for the rights of the contract and also to determine the trading strategy the writer should use to invest this fee in the (𝐵,𝑆)-market in such a way as to ensure that the writer will be able to cover the option if it is exercised. The fee should be large enough that the writer can, with riskless investing, cover the option, but be small enough that the writer does not make an unfair (i.e., riskless) profit.

In [6], we noted reasons to include hereditary price structures to a (𝐵,𝑆)-market model and then introduced such a model using a functional differential equation to describe the dynamics of the bank account and a stochastic functional differential equation to describe those of the stock account. The paper then obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The importance of including hereditary price structure in the stock price dynamics was also recognized by other researchers in recent years (see, e.g., [7–14]).

In particular, [6] was one of the firsts that took into consideration hereditary structure in studying the pricing problem of European option. There the authors obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The two papers [7, 9] developed an explicit formula for pricing European options when the underlying stock price follows a nonlinear stochastic delay equation with fixed delays (resp., variable delays) in the drift and diffusion terms. The paper [8] computed the logarithmic utility of an insider when the financial market is modelled by a stochastic delay equation. There the author showed that, although the market does not allow free lunches and is complete, the insider can draw more from his wealth than the regular trader. The paper also offered an alternative to the anticipating delayed Black-Scholes formula, by proving stability of European call option proces when the delay coefficients approach the nondelayed ones. The paper [10] derived the infinite-dimensional Black-Scholes equation for the (𝐵,𝑆)-market, where the bank account evolves according to a linear (deterministic) functional differential equation and the stock dynamics is described by a very general nonlinear stochastic functional differential equation. A power series solution is also developed for the equation. Following the same model studied in [10], the work in [11] shows that under very mild conditions the pricing function is the unique viscosity solution of the infinite-dimensional Black-Scholes equation. A finite difference approximation scheme for the solution of the equation is developed and convergence result is also obtained. We mention here that option pricing problems were also considered by [12–14] for a financial market that is more restricted than those of [10, 11].

This paper considers the pricing of a European option using a (𝐵,𝑆)-market, such as those in [6], in which the stock price and the asset in the riskless bank account both have hereditary price structures. Under the smoothness assumption of the payoff function, it is shown that the pricing function is the unique classical solution of the infinite-dimensional Black-Scholes equation. A spectral approximation scheme is developed using the Fourier series expansion in the space 𝐶[−ℎ,0] for the Black-Scholes equation. It is also shown that the 𝑛th approximant resembles the celebrated classical Black-Scholes equation in finite dimensions (see, e.g., [4, 5]).

This paper is organized as follows. Section 2 summarizes the definitions and key results of [6] that will be used throughout this paper. The concepts of Fréchet derivative and extended Fréchet derivative are introduced in Section 3, along with results needed to make use of these derivatives. In Section 4, the results regarding the infinite-dimensional Black-Scholes equation and its corollary are restated from [6, 10]. Section 5 details the spectral approximate solution scheme for this equation. Section 6 is the paper's conclusion, followed by an appendix with the proof of Proposition 3.2.

2. The European Option Problem with Hereditary Price Structures

To describe the financial model with hereditary price structures, we start by defining our probability space. Let 0<ℎ<∞ be a fixed constant. This constant will be the length of the time window in which the hereditary information is contained. If 𝑎,𝑏∈ℜ with 𝑎<𝑏, denote the space of continuous functions 𝜙∶[𝑎,𝑏]→ℜ by 𝐶[𝑎,𝑏]. Define
𝐶+[][][]𝑎,𝑏={𝜙∈𝐶𝑎,𝑏∣𝜙(𝜃)≥0∀𝜃∈𝑎,𝑏}.(2.1)
Note that 𝐶[𝑎,𝑏] is a real separable Banach space equipped with the uniform topology defined by the sup-norm ‖𝜙‖=sup𝑡∈[𝑎,𝑏]|𝜙(𝑡)| and 𝐶+[𝑎,𝑏] is a closed subset of 𝐶[𝑎,𝑏]. Throughout the end of this paper, we let 𝐂=𝐶[−ℎ,0] and
𝐂+[]}={𝜙∈𝐂∣𝜙(𝜃)≥0∀𝜃∈−ℎ,0(2.2)
for simplicity. If 𝜓∈𝐶[−ℎ,∞) and 𝑡∈[0,∞), let 𝜓𝑡∈𝐂 be defined by 𝜓𝑡(𝜃)=𝜓(𝑡+𝜃), 𝜃∈[−ℎ,0].

Let Ω=𝐶[−ℎ,∞), the space of continuous functions 𝜔∶[−ℎ,∞)→ℜ, and let 𝐹=𝐵(𝐶[−ℎ,∞)), the Borel 𝜎-algebra of subsets of 𝐶[−ℎ,∞) under the topology defined by the metric 𝑑∶Ω×Ω→ℜ, where𝑑𝜔,𝜔=∞𝑛=112𝑛sup−ℎ≤𝑡≤𝑛||𝜔(𝑡)−𝜔||(𝑡)1+sup−ℎ≤𝑡≤𝑛||𝜔(𝑡)−𝜔||.(𝑡)(2.3)

Let 𝐏 be the Wiener measure defined on (Ω,𝐹) with
[]𝐏{𝜔∈Ω∣𝜔(𝜃)=0∀𝜃∈−ℎ,0}=1.(2.4)
Note that the probability space (Ω,𝐹,𝐏) is the canonical Wiener space under which the coordinate maps 𝑊={𝑊(𝑡),𝑡≥0}, 𝑊(𝑡)∶𝐶[−ℎ,∞)→ℜ, defined by 𝑊(𝑡)(𝜔)=𝜔(𝑡) for all 𝑡≥−ℎ and 𝜔∈Ω is a standard Brownian motion and 𝐏{𝑊0=0}=1. Let the filtration 𝐹𝑊={𝐹(𝑡),𝑡≥−ℎ} be the 𝐏-augmentation of the natural filtration of the Brownian motion 𝑊, defined by 𝐹(𝑡)={∅,Ω} for all 𝑡∈[−ℎ,0] and
𝐹(𝑡)=𝜎(𝑊(𝑠),0≤𝑠≤𝑡),𝑡≥0.(2.5)

Equivalently, 𝐹(𝑡) is the smallest sub-𝜎-algebra of subsets of Ω with respect to which the mappings 𝑊(𝑠)∶Ω→ℜ are measurable for all 0≤𝑠≤𝑡. It is clear that the filtration 𝐹𝑊 defined above is right continuous in the sense of [15].

Consider the 𝐂-valued process {𝑊𝑡,𝑡≥0}, where 𝑊0=0 and 𝑊𝑡(𝜃)=𝑊(𝑡+𝜃), 𝜃∈[−ℎ,0] for all 𝑡≥0. That is, for each 𝑡≥0, 𝑊𝑡(𝜔)=𝜔𝑡 and 𝑊0=0. In [10], it is shown that 𝐹0=𝐹(𝑡) for 𝑡∈[−ℎ,0] and 𝐹(𝑡)=𝐹𝑡 for 𝑡≥0, where
𝐹𝑡𝑊=𝜎𝑠,0≤𝑠≤𝑡,𝑡≥0.(2.6)

The new model for the (𝐵,𝑆)-market introduced in [6] has a hereditary price structure in the sense that the rate of change of the unit price of the investor's assets in the bank account 𝐵 and that of the stock account 𝑆 depend not only on the current unit price but also on their historical prices. Specifically, we assume that 𝐵 and 𝑆 evolve according to the following two linear functional differential equations:
𝐵𝑑𝐵(𝑡)=𝐿𝑡𝑆𝑑𝑡,𝑡≥0,(2.7)𝑑𝑆(𝑡)=𝑀𝑡𝑆𝑑𝑡+𝑁𝑡𝑑𝑊(𝑡),𝑡≥0,(2.8)
with initial price functions 𝐵0=𝜙 and 𝑆0=𝜓, where 𝜙 and 𝜓 are given functions in 𝐂+. In the model, 𝐿, 𝑀, and 𝑁 are bounded linear functionals on the real Banach space 𝐂. The bounded linear functionals 𝐿,𝑀,𝑁∶𝐂→ℜ can be represented as (see [6])
𝐿(𝜙)=0−ℎ𝜙(𝜃)𝑑𝜂(𝜃),𝑀(𝜙)=0−ℎ𝑁𝜙(𝜃)𝑑𝜉(𝜃),(2.9)(𝜙)=0−ℎ𝜙(𝜃)𝑑𝜁(𝜃),𝜙∈𝐂,(2.10)
where the above integrals are to be interpreted as Lebesgue-Stieltjes integrals and 𝜂, 𝜉, and 𝜁 are functions that are assumed to satisfy the following conditions.

Assumption 2.1. The functions 𝜂,𝜉∶[−ℎ,0]→ℜ, are nondecreasing functions on [−ℎ,0] such that 𝜂(0)−𝜂(−ℎ)>0 and 𝜉(0)−𝜉(−ℎ)>0, and the function 𝜁∶[−ℎ,0]→ℜ is a function of bounded variation on [−ℎ,0] such that ∫0−ℎ𝜙(𝜃)𝑑𝜁(𝜃)≥𝜎>0 for every 𝜙∈𝐂+.

We will, throughout the end, extend the domain of the above three functions to 𝑅 by defining 𝜂(𝜃)=𝜂(−ℎ) for 𝜃≤−ℎ and 𝜂(𝜃)=𝜂(0) for 𝜃≥0, and so forth.

Proposition 2.3 in [6] provides an existence and uniqueness result under mild conditions, so the model makes sense mathematically to use. Note that the equations described by (2.7)-(2.8) include (1.1) as a special case. Therefore, the model considered in this paper is a generalization of that considered in most of the existing literature (see, e.g., [5]).

For the purpose of analyzing the discount rate for the bank account, let us assume that the solution process 𝐵(𝐿;𝜙)={𝐵(𝑡),−ℎ≤𝑡<∞} of (2.7) with the initial function 𝜙∈𝐂+ takes the following form:
𝐵(𝑡)=𝜙(0)𝑒𝑟𝑡,𝑡≥0,(2.11)
and 𝐵0=𝜙∈𝐂+. Then the constant 𝑟 satisfies the following equation:
𝑟=0−ℎ𝑒𝑟𝜃𝑑𝜂(𝜃).(2.12)
The existence and uniqueness of a positive number 𝑟 that satisfies the above equation is shown in [6].

Throughout the end, we will fix the initial unit price functions 𝜙, and 𝜓∈𝐂+, and the functional 𝑁∶𝐂→ℜ for the stock price described in (2.8) and (2.10). For the purpose of making the distinction when we interchange the usage of 𝑀∶𝐂→ℜ and 𝐿∶𝐂→ℜ in (2.8), we write the stock price process 𝑆(𝑀,𝑁;𝜓) as 𝑆(𝑀)={𝑆(𝑡),𝑡≥−ℎ} for simplicity. And, when the functional 𝐾∶𝐂→ℜ, 𝐾(𝜙𝑡)=𝑟𝜙(𝑡) is used in place of 𝑀∶𝐂→ℜ in (2.8), its solution process will be written as 𝑆(𝐾)={𝑆(𝑡),𝑡≥−ℎ}.

In [6], the basic theory of European option pricing using the (𝐵,𝑆)-market model described in (2.7)-(2.8) is developed. We summarize the key definitions and results below.

A trading strategy in the (𝐵,𝑆)-market is a progressively measurable vector process 𝜋={(𝜋1(𝑡),𝜋2(𝑡)),0≤𝑡<∞} defined on (Ω,𝐹,𝐏;𝐹𝑊) such that for each 𝑎>0,
𝑎0𝐄𝜋2𝑖(𝑡)𝑑𝑡<∞,𝑖=1,2,(2.13)
where 𝜋1(𝑡) and 𝜋2(𝑡) represent, respectively, the number of units of the bank account and the number of shares of the stock owned by the writer at time 𝑡≥0, and 𝐄 is the expectation with respect to 𝐏.

The writer's total asset is described by the wealth process 𝑋𝜋(𝑀)={𝑋𝜋(𝑡),0≤𝑡<∞} defined by
𝑋𝜋(𝑡)=𝜋1(𝑡)𝐵(𝑡)+𝜋2(𝑡)𝑆(𝑡),0≤𝑡<∞,(2.14)
where again 𝐵(𝐿;𝜙) and 𝑆(𝑀,𝑁;𝜓) are, respectively, the unit price of the bank account and the stock described in (2.7) and (2.8). This wealth process can clearly take both positive and negative values, since it is permissible that (𝜋1(𝑡),𝜋2(𝑡))∈ℜ2.

We will make the following basic assumption throughout this paper.

Assumption 2.2 (self-financing condition). In the (𝐵,𝑆)-market, it is assumed that all trading strategies 𝜋 satisfy the following self-financing condition:
𝑋𝜋(𝑡)=𝑋𝜋(0)+𝑡0𝜋1(𝑠)𝑑𝐵(𝑠)+𝑡0𝜋2(𝑠)𝑑𝑆(𝑠),0≤𝑡<∞,a.s.(2.15)
or equivalently,
𝑑𝑋𝜋(𝑡)=𝜋1(𝑡)𝑑𝐵(𝑡)+𝜋2(𝑡)𝑑𝑆(𝑡),0≤𝑡<∞.(2.16)

Using the same notation as in [6] (see also [10]) the set of all self-financing trading strategies 𝜋 will be denoted by SF(𝐿,𝑀,𝑁;𝜙,𝜓) or simply SF if there is no danger of ambiguity.

For the unit price of the bank account 𝐵(𝐿;𝜙)={𝐵(𝑡),𝑡≥0} and the stock 𝑆(𝑀,𝑁;𝜓)={𝑆(𝑡),𝑡≥0} described in (2.7) and (2.8), define
𝑊(𝑡)=𝑊(𝑡)+𝑡0𝛾𝐵𝑠,𝑆𝑠𝑑𝑠,𝑡≥0,(2.17)
where 𝛾∶𝐂+×𝐂+→ℜ is defined by
𝛾(𝜙,𝜓)=𝜙(0)𝑀(𝜓)−𝜓(0)𝐿(𝜙).𝜙(0)𝑁(𝜓)(2.18)

Define the process 𝑍(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑍(𝑡),𝑡≥0} by
𝑍(𝑡)=exp𝑡0𝛾𝐵𝑠,𝑆𝑠1𝑑𝑊(𝑠)−2𝑡0||𝛾𝐵𝑠,𝑆𝑠||2𝑑𝑠,𝑡≥0.(2.19)
The following results are proven in [6].

Lemma 2.3. The process 𝑍(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑍(𝑡),𝑡≥0} defined by (2.19) is a martingale defined on (Ω,𝐹,𝐏;𝐹𝑊).

Lemma 2.4. There exists a unique probability measure 𝐏 defined on the canonical measurable space (Ω,𝐹) such that
𝟏𝐏(𝐴)=𝐄𝐴𝑍(𝑇)∀𝐴∈𝐹𝑇,0<𝑇<∞,(2.20)
where 𝟏𝐴 is the indicator function of 𝐴∈𝐹𝑇.

Lemma 2.5. The process 𝑊 defined by (2.17) is a standard Brownian motion defined on the filtered probability space (Ω,𝐹,𝐏;𝐹𝑊).

From the above, it has been shown (see [6, equation (14)]) that
𝑆𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑡𝑑𝑊(𝑡),(2.21)
with 𝑆0=𝜓∈𝐂+. It is also clear that the probabilistic behavior of 𝑆(𝑀) under the probability measure 𝐏 is the same as that of 𝑆(𝐾) under the probability measure 𝐏; that is, they have the same distribution.

Define the process 𝑌𝜋(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑌𝜋(𝑡),𝑡≥0}, called the discounted wealth process, by
𝑌𝜋𝑋(𝑡)=𝜋(𝑡)𝐵(𝑡),𝑡≥0.(2.22)
We say that a trading strategy 𝜋 from SF(𝐿,𝑀,𝑁;𝜙,𝜓) belongs to a subclass SF𝜍⊂SF if 𝐏 a.s.
𝑌𝜋𝐄(𝑡)≥−𝜍∣𝐹𝑡,𝑡≥0,(2.23)
where 𝐄 is the expectation with respect to 𝐏, 𝜍 is a nonnegative 𝐹-measurable random variable such that 𝐄[𝜍]<∞. We say that 𝜋 belongs to SF+⊂SF if 𝜍≥0.

In [6, 10], it is shown that 𝑌𝜋 is a local martingale; for 𝜋∈SF𝜍, 𝑌𝜋 is a supermartingale, and is a nonnegative supermartingale if 𝜋∈SF+.

Throughout, we assume the reward function Λ is an 𝐹𝑇-measurable nonnegative random variable satisfying the following condition:
𝐄Λ1+𝜖<∞,(2.24)
for some 𝜖>0. Here, 𝑇>0 is the expiration time. (Note that the above condition on Λ implies that 𝐄[Λ]<∞.)

Let Λ be a nonnegative 𝐹𝑇-measurable random variable satisfying (2.24). A trading strategy 𝜋∈SF is a (𝑀;Λ,𝑥)-hedge of European type if
𝑋𝜋(0)=𝜋1(0)𝜙(0)+𝜋2(0)𝜓(0)=𝑥(2.25)
and 𝐏 a.s.
𝑋𝜋(𝑇)≥Λ.(2.26)
We say that a (𝑀;Λ,𝑥)-hedge trading strategy 𝜋∗∈SF(𝑀) is minimal if
𝑋𝜋(𝑇)≥𝑋𝜋∗(𝑇)(2.27)
for any (𝑀;Λ,𝑥)-hedge strategy 𝜋∈SF(𝑀).

Let Π(𝑀;Λ,𝑥) be the set of (𝑀;Λ,𝑥)-hedge strategies from SF+(𝑀). Define
𝐶(𝑀;Λ)=inf{𝑥≥0∶Π(𝑀;Λ,𝑥)≠∅}.(2.28)
The value 𝐶(𝑀;Λ) defined above is called the rational price of the contingent claim of European type. If the infimum in (2.28) is achieved, then 𝐶(𝑀;Λ) is the minimal possible initial capital for which there exists a trading strategy 𝜋∈SF+(𝑀) possessing the property that 𝐏 a.s. 𝑋𝜋(𝑇)≥Λ.

Let 𝑌(𝑀)={𝑌(𝑡),0≤𝑡≤𝑇} be defined by
𝐄Λ𝑌(𝑡)=∣𝐹𝐵(𝑇)𝑡,0≤𝑡≤𝑇,(2.29)
where 𝐹𝑡𝑊=𝜎(𝑠,0≤𝑠≤𝑡). In [10], it is shown that the process 𝑌(𝑀) is a martingale defined on 𝑊)(Ω,𝐹,𝐏;𝐹 and can be represented by
𝑌(𝑡)=𝑌(0)+𝑡0𝛽(𝑠)𝑑𝑊(𝑠),(2.30)
where 𝛽={𝛽(𝑡),0≤𝑡≤𝑇} that is 𝐹𝑊-adapted and ∫𝑇0𝛽2(𝑡)𝑑𝑡<∞ (𝐏 a.s.).

The following lemma and theorem provide the main results of [6, 10]. Let 𝜋∗={(𝜋∗1(𝑡),𝜋∗2(𝑡)),0≤𝑡≤𝑇} be a trading strategy, where
𝜋∗2(𝑡)=𝛽(𝑡)𝐵(𝑡)𝑁𝑆𝑡,𝜋∗1𝑆(𝑡)=𝑌(𝑡)−(𝑡)𝜋𝐵(𝑡)∗2[].(𝑡),𝑡∈0,𝑇(2.31)

Lemma 2.6. 𝜋∗∈SF(𝑀) and for each 𝑡∈[0,𝑇], 𝑌(𝑡)=𝑌𝜋∗(𝑡) for each 𝑡∈[0,𝑇] where again 𝑌𝜋∗ is the process defined in (2.22) with the minimal strategy 𝜋∗ defined in (2.31).

Theorem 2.7. Let Λ be an 𝐹𝑇-measurable random variable defined on the filtered probability space (Ω,𝐹,𝐏;𝐹𝑊) that satisfies (2.24). Then the rational price 𝐶(𝑀;Λ) defined in (2.28) is given by
𝐄𝑒𝐶(𝑀;Λ)=−𝑟𝑇Λ,(2.32)
where 𝑟 is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge 𝜋∗={(𝜋∗1(𝑡),𝜋∗2(𝑡)),0≤𝑡≤𝑇}, where
𝜋∗2(𝑡)=𝛽(𝑡)𝐵(𝑡)𝑁𝑆𝑡,𝜋∗1(𝑡)=𝑌𝜋∗(𝑡)−𝜋∗2𝑆(𝑡)(𝑡),𝐵(𝑡)(2.33)
and the process 𝛽={𝛽(𝑡),0≤𝑡≤𝑇} is given by (2.30). If in addition, the reward Λ is intrinsic, that is, Λ=Γ(𝑆(𝑀)) for some measurable function Γ∶𝐂+→ℜ, then the rational price 𝐶(𝑀;Λ) does not depend on the mean growth rate 𝑀 of the stock and
𝐄𝑒𝐶(Λ)=−𝑟𝑇Λ.(2.34)

3. Fréchet and Extended Fréchet Derivatives

In this section, results are proven that allow the use of a Dynkins formula for stochastic functional differential equation as found in [16, 17]. We assume contingent claims of European type in which the 𝐹𝑇-measurable reward function Λ has the explicit expression Λ=𝑓(𝑆𝑇), where again 𝑆𝑇(𝜃)=𝑆(𝑇+𝜃), 𝜃∈[−ℎ,0] and 𝑆(𝐾)={𝑆(𝑡),𝑡≥0} is the unit price of the stock described by the following equation:
𝑆𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑡𝑑𝑊(𝑡),𝑡≥0,(3.1)
where 𝑆0=𝜓∈𝐂+. Throughout this section, we assume that 𝑆(𝑡), and therefore 𝑁(𝑆𝑡), are uniformly bounded almost surely. This assumption is realistic for the price of a stock during time interval [0,𝑇] in a financial system with finite total wealth.

The remaining sections make extensive use of Fréchet derivatives. Let 𝐂∗ be the space of bounded linear functionals Φ∶𝐂→ℜ. 𝐂∗ is a real separable Banach space under the supremum operator norm
‖Φ‖=sup𝜙≠0||||Φ(𝜙).‖𝜙‖(3.2)
For Ψ∶[0,𝑇]×𝐂→ℜ, we denote the Fréchet derivative of Ψ at 𝜙∈𝐂 by 𝐷Ψ(𝑡,𝜙). The second Fréchet derivative at 𝜙 is denoted as 𝐷2Ψ(𝑡,𝜙).

Let Γ be the vector space of all simple functions of the form 𝑣𝟏{0}, where 𝑣∈𝑅 and 𝟏{0}∶[−ℎ,0]→ℜ is defined by
𝟏{0}[(𝜃)=0,for𝜃∈−ℎ,0),1,for𝜃=0.(3.3)
Form the direct sum 𝐂⊕Γ and equip it with the complete norm
‖‖𝜙+𝑣𝟏{0}‖‖=sup𝜃∈[−ℎ,0]||||𝜙(𝜃)+|𝑣|,𝜙∈𝐂,𝑣∈ℜ.(3.4)
Then 𝐷Ψ(𝑡,𝜙) has a unique continuous linear extension from 𝐂⊕Γ to ℜ which we will denote by 𝐷Ψ(𝑡,𝜙), and similarly for 𝐷2Ψ(𝑡,𝜙); see [16] or [17] for more details.

Finally, we define
𝐆(Ψ)𝑡,𝜓𝑡=lim𝑢→0+1𝑢Ψ𝑡,𝜓𝑡+𝑢−Ψ𝑡,𝜓𝑡(3.5)
for all 𝑡∈[0,∞) and 𝜓∈𝐂+, where 𝜓∶[−ℎ,∞)→ℜ is defined by
[𝜓(𝑡)=𝜓(𝑡)if𝑡∈−ℎ,0)𝜓(0)if𝑡≥0.(3.6)
Let 𝑓∶𝐂→ℜ. We say that 𝑓∈𝐶1(𝐂) if 𝑓 has a continuous Fréchet derivative. Similarly, 𝑓∈𝐶𝑛(𝐂) if 𝑓 has a continuous 𝑛th Fréchet derivative. For 𝑓∶𝑅+×𝐂→ℜ, we say that 𝑓∈𝐶∞,𝑛([0,∞)×𝐂) if 𝑓 is infinitely differentiable in its first variable and has a continuous 𝑛th partial derivative in its second variable.

Proof. That 𝑒−𝑟(𝑇−𝑡) is 𝐶∞[0,∞) is clear, so we have only to show that Υ∈𝐶2(𝐂), where Υ(𝜑)=𝐄[𝑓(𝑆𝑇)∣𝑆𝑡=𝜑] given that 𝑓∈𝐶2(𝐂).We have that
𝑆𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑡𝑑𝑊(𝑡),𝑡≥0,(3.8)
with 𝑆0=𝜓∈𝐂+. Under Assumption 2.1 on 𝑁∶𝐂→ℜ and the properties of Υ, it can be shown that there exists 𝐻∶ℜ×ℜ×𝐂→𝐂 such that 𝑆𝑡=𝐻(𝑡,𝑊(𝑡),𝜓). Therefore,
𝐄𝑓𝑆Υ(𝜑)=𝑇∣𝑆𝑡=1=𝜑√2𝜋∞−∞𝑓(𝐻(𝑇−𝑡,𝑦,𝜑))𝑒−𝑦2/2𝑑𝑦.(3.9)
By Theorem 3.2, Chapter 2 of [16], 𝐻(𝑡,𝑦,⋅)∈𝐶1(𝐂) as a function of 𝜓. By a second application of the same theorem (since 𝑓∈𝐶2(𝐂)), we have that 𝐻(𝑡,𝑦,⋅)∈𝐶2(𝐂) as a function of 𝜓. Define 𝑔∶ℜ×ℜ×𝐂→ℜ by 𝑔=𝑓∘𝐻. Since 𝑓∈𝐶2(𝐂) and 𝐻(𝑡,𝑦,⋅)∈𝐶2(𝐂) in its third variable, 𝑔(𝑡,𝑦,⋅)∈𝐶2(𝐂). Hence, for 𝜑,𝜙∈𝐂,
𝐄𝑓𝑆𝑇∣𝑆𝑡−𝐄𝑓𝑆=𝜑+𝜙𝑇∣𝑆𝑡=1=𝜑√2𝜋∞−∞[]𝑒𝑓(𝐻(𝑇−𝑡,𝑦,𝜑+𝜙))−𝑓(𝐻(𝑇−𝑡,𝑦,𝜑))−𝑦2/2=1𝑑𝑦√2𝜋∞−∞𝐷𝑔(𝑇−𝑡,𝑦,𝜑)(𝜙)𝑒−𝑦2/21𝑑𝑦+√2𝜋∞−∞𝑜(𝜙)𝑒−𝑦2/2𝑑𝑦,(3.10)
where 𝑜(𝜙) is a function mapping continuous functions into the reals such that
𝑜(𝜙)‖𝜙‖⟶0as‖𝜙‖⟶0.(3.11)
The last integral is clearly 𝑜(𝜙) and
1√2𝜋∞−∞𝐷𝑔(𝑇−𝑡,𝑦,𝜑)(𝜙)𝑒−𝑦2/2𝑑𝑦(3.12)
is bounded and linear in 𝜙, so this integral is the first Fréchet derivative with respect to 𝜑. Since 𝑔(𝑡,𝑦,⋅)∈𝐶2(𝐂), the process can be repeated, giving a second Fréchet derivative with respect to 𝜑 and so Υ∈𝐶2(𝐂).

Proposition 3.2. Let 𝜑∈𝐂 and 𝑓∶𝐂→ℜ. Further assume 𝑓∈𝐶2(𝐂) and let Ψ∶[0,𝑇]×𝐂→ℜ be defined by Ψ(𝑡,𝜑)=𝑒−𝑟(𝑇−𝑡)𝐸𝑓𝑆𝑇∣𝑆𝑡.=𝜑(3.13)
Then if 𝐷𝑓 and 𝐷2𝑓 are globally Lipschitz, then so is 𝐷2Ψ.

Recall from Proposition 3.1 that 𝑔∶𝑅×𝑅×𝐂→𝑅 is 𝑓∘𝐻 where 𝑆𝑡=𝐻(𝑡,𝑊(𝑡),𝜓) with 𝑆0=𝜓∈𝐂+.

Proposition 3.3. Let 𝜑∈𝐂 and 𝑓∶𝐂→ℜ. Further assume 𝑓∈𝐶2(𝐂) and let Ψ∶[0,𝑇]×𝐂→ℜ be defined by
Ψ(𝑡,𝜑)=𝑒−𝑟(𝑇−𝑡)𝐸𝑓𝑆𝑇∣𝑆𝑡.=𝜑(3.14)
Then if 𝑓 and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓𝑡) are globally bounded, then so is 𝐆(Ψ)(𝑡,𝜓𝑡).

Proof. We have that
𝐆(Ψ)𝑡,𝜓𝑡=lim𝑢→0+1𝑢Ψ𝑡,𝜓𝑡+𝑢−Ψ𝑡,𝜓𝑡=lim𝑢→0+1𝑢1√2𝜋∞−∞𝑔𝑇−𝑡,𝑦,𝜓𝑡+𝑢−𝑔𝑇−𝑡,𝑦,𝜓𝑡𝑒−𝑦2/2=1𝑑𝑦√2𝜋∞−∞lim𝑢→0+1𝑢𝑔𝑇−𝑡,𝑦,𝜓𝑡+𝑢−𝑔𝑇−𝑡,𝑦,𝜓𝑡𝑒−𝑦2/2≤1𝑑𝑦√2𝜋∞−∞𝑀𝑒−𝑦2/2𝑑𝑦=𝑀<∞,(3.15)
where we used the assumption that 𝑓 and hence 𝑔 are globally bounded to move the limit inside the integral and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓𝑡)≤𝑀<∞.

Remark 3.4. Note that since 𝐆(Ψ)(𝜓𝑠) is bounded for all 𝑠∈[0,𝑇], ∫𝑡0𝐷Ψ(𝑠,𝜓𝑠)(𝑑𝜓𝑠) exits. Also, if 𝐷2𝑓 is bounded, ∫𝑡0𝐷2Ψ(𝑠,𝜓𝑠)(𝑑𝜓𝑠,𝑑𝜓𝑠) exits (see [18]).

4. The Infinite-Dimensional Black-Scholes Equation

It is known (e.g., [4, 5]) that the classical Black-Scholes equation is a deterministic parabolic partial differential equation (with a suitable auxiliary condition) the solution of which gives the value of the European option contract at a given time. Propositions 3.1 through 3.3 allow us to use the Dynkin formula in [16]. With it, a generalized version of the classical Black-Scholes equation can be derived for when the (𝐵,𝑆)-market model is (2.7) and (2.8). The following theorem is a restatement of Theorem 3.1 in [10].

Theorem 4.1. Let Ψ(𝑡,𝜑)=𝑒−𝑟(𝑇−𝑡)𝐄[𝑓(𝑆𝑇)∣𝑆𝑡=𝜑], where 𝑆0=𝜓∈𝐂+ and 𝑡∈[0,𝑇]. Let 𝑓 be a 𝐶2(𝐂) function with 𝐷𝑓 and 𝐷2𝑓 globally Lipschitz and let Λ=𝑓(𝑆𝑇) and 𝑥=𝑋𝜋∗(0). Finally, let 𝑓 and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓𝑡) be globally bounded. Then if 𝑋𝜋∗(𝑡)=Ψ(𝑡,𝑆𝑡) is the wealth process for the minimal (Λ,𝑥)-hedge, one has
𝜕𝑟Ψ(𝑡,𝜑)=𝜕𝑡Ψ(𝑡,𝜑)+𝐆(Ψ)𝑡,𝜑𝑡+𝐷Ψ(𝑡,𝜑)𝑟𝜑(0)𝟏{0}+12𝐷2Ψ(𝑡,𝜑)𝑁(𝜑)𝟏{0},𝑁(𝜑)𝟏{0}[,a.s.∀(𝑡,𝜑)∈0,𝑇)×𝐂+,(4.1)
where
Ψ(𝑇,𝜑)=𝑓(𝜑)∀𝜑∈𝐂+,(4.2)
and the trading strategy (𝜋∗1(𝑡),𝜋∗2(𝑡)) is defined by
𝜋∗2(𝑡)=𝟏𝐷Ψ(𝑡,𝜑){0}𝜋a.s.,∗11(𝑡)=𝑋𝐵(𝑡)𝜋∗(𝑡)−𝜑(0)𝜋∗2.(𝑡)(4.3)
Furthermore, if (4.1) and (4.2) hold, then Ψ(𝑡,𝑆𝑡) is the wealth process for the (Λ,𝑥)-hedge with 𝜋∗2(𝑡)=𝐷Ψ(𝑡,𝑆𝑡)(𝟏{0}) and 𝜋∗1(𝑡)=(1/𝐵(𝑡))[𝑋𝜋∗(𝑡)−𝑆(𝑡)𝜋∗2(𝑡)].

Proof. The theorem is a restatement of Theorem 3.1 in [10] and is therefore omitted.

NoteEquations (4.1) and (4.2) are the generalized Black-Scholes equation for the (𝐵,𝑆)-market with hereditary price structure as described by (2.7) and (2.8).

5. Approximation of Solutions

In this section, we will solve the generalized Black-Scholes equation (4.1)-(4.2) by considering a sequence of approximations of its solution. By a (classical) solution to (4.1)-(4.2), we mean Ψ∶[0,𝑇]×𝐂→ℜ satisfying the following conditions:

(i)Ψ∈𝐶1,2([0,𝑇]×𝐂),(ii)Ψ(𝑇,𝜑)=𝑓(𝜑) for all 𝜑∈𝐂,(iii)Ψ satisfies (4.1).

The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of 𝐂, solving (4.1)-(4.2) on these subspaces, and then showing that as the dimension of the subspaces goes to infinity, the finite-dimensional solutions converge to a solution of the original problem. Theorem 5.2, Remark 5.3, and Corollary 5.4 show that the generalized Black-Scholes equation can be solved by solving two simpler equations. The first of these, a first-order partial differential equation, can be handled by traditional techniques once the second equation is solved. Theorem 5.5 provides a solution to the second. Proposition 5.7, which uses Lemma 5.6, gives a generalized Black-Scholes formula for the standard European call option when used in conjunction with Theorem 5.2.

We start by noting that 𝐂⊂𝐿2[−ℎ,0] where 𝐿2[−ℎ,0] is the space of all square-integrable functions on the interval [−ℎ,0]. Furthermore, 𝐂 is dense in 𝐿2[−ℎ,0]. It is well known (e.g., [19]) that even extensions of a function 𝜑 in 𝐿2[−ℎ,0] may be represented by a cosine Fourier series where
‖‖‖‖𝜑−𝑁𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖2⟶0(5.1)
as 𝑁→∞ where
𝑎0=1ℎ0−ℎ𝑎𝜑(𝜃)𝑑𝜃,𝑖=2ℎ0−ℎ𝜑(𝜃)cos2𝜋𝑖𝜃ℎ𝑑𝜃,𝑖=1,2,3,….(5.2)
Here,
‖𝑓‖22=0−ℎ𝑓2(𝜃)𝑑𝜃(5.3)
for 𝑓∈𝐿2[−ℎ,0]. If 𝜑 is Hölder-continuous, then the convergence is also point wise (see, e.g., [20]).

Throughout this section, we let 𝐿2𝑛[−ℎ,0] be the subspace of 𝐿2[−ℎ,0] consisting of functions that can be represented as a finite Fourier series, that is, 𝜑(𝑛)∈𝐿2𝑛[−ℎ,0] if
𝜑(𝑛)(𝜃)=𝑛𝑖=0𝑎𝑖cos2𝜋𝑖𝜃ℎ(5.4)
for all 𝜃∈[−ℎ,0].

We will see that it is convenient having a spanning set {𝑓𝑖}∞𝑖=0 for 𝐿2𝑛[−ℎ,0] where 𝑓𝑖∶[−ℎ,0]→ℜ for 𝑖=0,1,… such that 𝑓𝑖(0)=1 for all 𝑖 and 𝑁(𝑓𝑖∫)=0−ℎ𝑓𝑖(𝜃)𝑑𝜁(𝜃)=𝛿 for all 𝑖. Here, ∫𝛿=0−ℎ𝑑𝜁(𝜃)∈ℜ. Let 𝑞∶[−ℎ,0]→ℜ be any function such that 𝑁(𝑞)=𝑞(0)≠0. For example, let
𝑞(𝜃)=1+1−𝛿−𝑑2𝑑1𝜃+𝜃2,(5.5)
where 𝑑1=∫0−ℎ𝜃𝑑𝜁(𝜃) and 𝑑2=∫0−ℎ𝜃2𝑑𝜁(𝜃). To this end, we define the following functions. Let
𝑓0[],𝑓(𝜃)=1∀𝜃∈−ℎ,01(𝜃)=𝛼1,1+𝛼1,2𝑞[],𝑓(𝜃)∀𝜃∈−ℎ,02(𝜃)=𝛼2,1𝑞(𝜃)+𝛼2,2cos2𝜋𝜃ℎ[],∀𝜃∈−ℎ,0(5.6)
and for 𝑖=3,4,…,
𝑓𝑖(𝜃)=𝛼𝑖,1cos2𝜋(𝑖−2)𝜃ℎ+𝛼𝑖,2cos2𝜋(𝑖−1)𝜃ℎ[]∀𝜃∈−ℎ,0.(5.7)
Recall that 𝑁∶𝐿2[−ℎ,0]→ℜ is defined by
𝑁(𝜑)=0−ℎ𝜑(𝜃)𝑑𝜁(𝜃),(5.8)
and let
𝑐𝑖=𝑁cos2𝜋𝑖⋅ℎ=0−ℎcos2𝜋𝑖𝜃ℎ𝑑𝜁(𝜃).(5.9)
Here again 𝑞∶[−ℎ,0]→ℜ is any function such that 𝑁(𝑞)=𝑞(0)≠0. For example, 𝑞 can be chosen as in (5.5). In this case, the constant 𝛼1,2 is nonzero but otherwise arbitrary,
𝛼1,1=1−𝛼1,2𝛼𝑞(0),2,1=𝛿−𝑐1𝑞(0)1−𝑐1,𝛼2,2=1−𝛼2,1𝑞(0),(5.10)
and so on with
𝛼𝑖,2=𝛿−𝑐𝑖−2𝑐𝑖−1−𝑐𝑖−2,𝛼𝑖,1=1−𝛼𝑖,2(5.11)
for 𝑖≥3.

Lemma 5.1. The set {𝑓𝑖}∞𝑖=0 defined in (5.6) and (5.7) forms a spanning set for 𝐿2[−ℎ,0] in the sense that
‖‖‖‖𝜑−𝑛+1𝑖=0𝑥𝑖𝑓𝑖‖‖‖‖2⟶0(5.12)
as 𝑛→∞, where the 𝑥𝑖 are defined by
𝑥𝑛+1=𝑎𝑛𝛼𝑛+1,2,𝑥𝑛=𝑎𝑛−1−𝑥𝑛+1𝛼𝑛+1,1𝛼𝑛,2,(5.13)
and continuing using
𝑥𝑖=𝑎𝑖−1−𝑥𝑖+1𝛼𝑖+1,1𝛼𝑖,2(5.14)
until
𝑥1𝑥=−2𝛼2,1𝛼1,2,𝑥0=𝑎0−𝑥1𝛼1,1.(5.15)
This set of functions has the properties that 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿 for all 𝑖=0,1,….

Proof. For any 𝜑∈𝐿2[−ℎ,0], we can construct an even extension 𝜙∈𝐿2[−ℎ,ℎ] where 𝜙(𝜃)=𝜑(𝜃) for all 𝜃∈[−ℎ,0] and 𝜙(𝜃)=𝜑(−𝜃) for all 𝜃∈[0,ℎ]. The function 𝜙 may be represented by a Fourier series of cosine functions
𝜙(𝜃)∼𝑁𝑖=0𝑎𝑖cos2𝜋𝑖𝜃ℎ,(5.16)
where the “∼’’ is used to indicate that
‖‖‖‖𝜙−𝑁𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖2⟶0(5.17)
as 𝑁→∞. In what mentioned before,
𝑎0=1ℎ0−ℎ𝑎𝜙(𝜃)𝑑𝜃,𝑖=2ℎ0−ℎ𝜙(𝜃)cos2𝜋𝑖𝜃ℎ𝑑𝜃(5.18)
for all 𝑖=1,2,…. For simplicity, we will replace the “∼’’ with an equality sign knowing that mean-square convergence is implied.For the Fourier series, the basis is
cos2𝜋𝑖𝜃ℎ∞𝑖=0,(5.19)
so the first term of this basis and {𝑓𝑖}∞𝑖=0 are the same, namely, the constant “1.’’ Clearly 𝑓0(0)=1 and 𝑁(𝑓0)=𝛿. The first part of this proof is to show that for all 𝑖=0,1,…, 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿.For 𝑓1, we have that 𝑓1(0)=𝛼1,1+𝛼1,2𝑞(0)=1 which implies that
𝛼1,1=1−𝛼1,2𝑞(0).(5.20)
Also, 𝑁(𝑓1)=𝛼1,1𝛿+𝛼1,2𝑁(𝑞)=𝛿. Since we do not want 𝛼1,2=0, we require that
𝑁(𝑞)=𝑞(0).(5.21)
There are no restrictions on 𝛼1,2 other than 𝛼1,2≠0.For 𝑓2, 𝛼2,1𝑞(0)+𝛼2,2=1 requires that
𝛼2,2=1−𝛼2,1𝑞(0).(5.22)
Since we want 𝛼2,1𝑁(𝑞)+𝛼2,2𝑐1=𝛿, then
𝛼2,1=𝛿−𝑐1𝑁(𝑞)−𝑞(0)𝑐1=𝛿−𝑐1𝑞(0)1−𝑐1.(5.23)The rest of the 𝑓𝑖, that is, where 𝑖≥3, are handled alike. In order that 𝑓𝑖(0)=1, we require that 𝛼𝑖,1=1−𝛼𝑖,2. To ensure that 𝑁(𝑓𝑖)=𝛿,
𝛼𝑖,2=𝛿−𝑐𝑖−2𝑐𝑖−1−𝑐𝑖−2.(5.24)
We have now shown that the sequence of functions {𝑓𝑖}∞𝑖=0 is such that 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿 for all 𝑖=0,1,…. Now it must be shown that this sequence is a spanning set for 𝐿2[−ℎ,0]. To do this, we will compare this sequence of functions with the cosine Fourier sequence of functions.Consider 𝜑(𝑛)∶[−ℎ,0]→ℜ where
𝜑(𝑛)(𝜃)=𝑛𝑖=0𝑎𝑖cos2𝜋𝑖𝜃ℎ.(5.25)
We would like
𝜑(𝑛)(𝜃)=𝑛+1𝑖=0𝑥𝑖𝑓𝑖(𝜃)(5.26)
for some set {𝑥𝑖}𝑛+1𝑖=0 of real numbers. By the Fourier expansion,
𝜑(𝑛)(𝜃)=𝑎0+𝑎1cos2𝜋𝜃ℎ+⋯+𝑎𝑛cos2𝜋𝑛𝜃ℎ.(5.27)
We want {𝑥𝑖}𝑛+1𝑖=0 where
𝜑(𝑛)(𝜃)=𝑥0+𝑥1𝛼1,1+𝛼1,2𝑞(𝜃)+𝑥2𝛼2,1𝑞(𝜃)+𝛼2,2cos2𝜋𝜃ℎ+𝑥3𝛼3,1cos2𝜋𝜃ℎ+𝛼3,2cos4𝜋𝜃ℎ+⋯+𝑥𝑛𝛼𝑛,1cos2𝜋(𝑛−2)𝜃ℎ+𝛼𝑛,2cos2𝜋(𝑛−1)𝜃ℎ+𝑥𝑛+1𝛼𝑛+1,1cos2𝜋(𝑛−1)𝜃ℎ+𝛼𝑛+1,2cos2𝜋𝑛𝜃ℎ=𝑥0+𝑥1𝛼1,1𝑥+𝑞(𝜃)1𝛼1,2+𝑥2𝛼2,1+cos2𝜋𝜃ℎ𝑥2𝛼2,2+𝑥3𝛼3,1+⋯+cos2𝜋𝑖𝜃ℎ𝑥𝑖+1𝛼𝑖+1,2+𝑥𝑖+2𝛼𝑖+2,1+⋯+cos2𝜋(𝑛−1)𝜃ℎ𝑥𝑛𝛼𝑛,2+𝑥𝑛+1𝛼𝑛+1,1+cos2𝜋𝑛𝜃ℎ𝑥𝑛+1𝛼𝑛+1,2.(5.28)
Equating the last coefficients gives
𝑥𝑛+1=𝑎𝑛𝛼𝑛+1,2,𝑥𝑛=𝑎𝑛−1−𝑥𝑛+1𝛼𝑛+1,1𝛼𝑛,2.(5.29)
Continuing,
𝑥𝑖=𝑎𝑖−1−𝑥𝑖+1𝛼𝑖+1,1𝛼𝑖,2,(5.30)
and finally
𝑥1𝑥=−2𝛼2,1𝛼1,2,𝑥0=𝑎0−𝑥1𝛼1,1.(5.31)
Hence, with the above choice of {𝑥𝑖}𝑛+1𝑖=0,
𝑛𝑖=0𝑎𝑖cos2𝜋𝑖𝜃ℎ=𝑛+1𝑖=0𝑥𝑖𝑓𝑖(𝜃),(5.32)
and so
‖‖‖‖𝜑−𝑛+1𝑖=0𝑥𝑖𝑓𝑖‖‖‖‖2=‖‖‖‖𝜑−𝑛+1𝑖=0𝑥𝑖𝑓𝑖+𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ−𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖2=‖‖‖‖𝜑−𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ+𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ−𝑛+1𝑖=0𝑥𝑖𝑓𝑖‖‖‖‖2≤‖‖‖‖𝜑−𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖2+‖‖‖‖𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ−𝑛+1𝑖=0𝑥𝑖𝑓𝑖‖‖‖‖2=‖‖‖‖𝜑−𝑛𝑖=0𝑎𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖2⟶0(5.33)
as 𝑛→∞.

To find an approximate solution to the generalized Black-Scholes equation we start by letting 𝑋𝜋∗(𝑡)=𝐄[𝑓(𝑆𝑇)∣𝑆𝑡=𝜑] (from [6]) and approximating 𝜑 by
𝜑(𝑛)=𝑛+1𝑖=0𝑥𝑖𝑓𝑖.(5.34)
We define the space 𝐂𝑛 as the set of all continuous functions that can be represented by this summation for some {𝑥𝑖}∞𝑖=0. Note that 𝐂𝑛⊂𝐿2𝑛[−ℎ,0]. Also define 𝑒𝑛∶ℜ𝑛+2→ℜ by
𝑒𝑛=⃗𝑥𝑛+1𝑖=0𝑥𝑖𝑓𝑖,(5.35)
so that Ψ(𝑡,𝜑(𝑛))=Ψ(𝑡,𝑒𝑛(⃗𝑥)). Define Ψ𝑛∶[0,𝑇]×ℜ𝑛+2→ℜ by Ψ𝑛(𝑡,⃗𝑥)=Ψ(𝑡,𝜑(𝑛)) provided that the ⃗𝑥 is formed by the coefficients of 𝜑(𝑛) in the spanning set {𝑓𝑖}∞𝑖=0. In general, ⃗𝑥(𝑡) is formed by the coefficients of 𝜑𝑡(𝑛) in the spanning set {𝑓𝑖}∞𝑖=0. Also, define 𝑣𝑛∶[−ℎ,0]→ℜ by
𝑣𝑛⎧⎪⎨⎪⎩1(𝜃)=0,for𝜃∈−ℎ,−𝑛,−1𝑛𝜃+1,for𝜃=𝑛.,0(5.36)
Last, let 𝑔𝑛∶[0,𝑇]×𝐂𝑛×ℜ𝑛+1×𝐶1,2([0,𝑇]×𝐂)→ℜ be defined by
𝑔𝑛𝑡,𝜑(𝑛),⃗𝑥,Ψ=𝑟𝑛+1𝑖=0𝑥𝑖𝐷Ψ(𝑡,𝜑(𝑛))𝟏{0}−𝑛+1𝑖=0𝑘𝑖𝜕𝜕𝑥𝑖Ψ𝑛+𝛿𝑡,⃗𝑥22𝑛+1𝑖=0𝑥𝑖2𝐷2Ψ(𝑡,𝜑(𝑛))𝟏{0},𝟏{0}−𝑛+1𝑖,𝑗=0𝑘𝑖𝑘𝑗𝜕2𝜕𝑥𝑖𝜕𝑥𝑗Ψ𝑛,𝑡,⃗𝑥(5.37)
where the 𝑘𝑖 are the coefficients of 𝑣𝑛 using the spanning set {𝑓𝑖}∞𝑖=0. Finally, define the operator (⋅)𝑛∶𝐂→𝐂𝑛 by
(𝜑)𝑛=𝑛+1𝑖=0𝑥𝑖𝑓𝑖,(5.38)
where the right-hand side is the first 𝑛+2 terms of the {𝑓𝑖}-expansion of 𝜑.

We are now ready for a theorem which enables us to approximate the solution of the infinite-dimensional Black-Scholes equation by solving a first-order real-valued partial differential equation and an equation similar to the generalized Black-Scholes equation but without the 𝐆(Ψ)(𝑡,𝜑𝑡) term. The lack of this term allows approximate solutions to be found using traditional techniques.

Theorem 5.2. Let 𝑆0=𝜓∈𝐂+ and 𝑡∈[0,𝑇]. Let 𝑓 be a 𝐶2(𝐂) function satisfying the conditions of Theorem 4.1 and let Λ=𝑓(𝑆𝑇). Then
𝑟Ψ𝑡,𝜑(𝑛)=𝜕Ψ𝜕𝑡𝑡,𝜑(𝑛)+𝐆(Ψ)𝑡,𝜑𝑡𝑛+𝐷Ψ𝑡,𝜑(𝑛)𝑟𝜑(𝑛)(0)𝟏{0}+12𝐷2Ψ𝑡,𝜑(𝑛)𝑁𝜑(𝑛)𝟏{0}𝜑,𝑁(𝑛)𝟏{0},∀𝑡,𝜑(𝑛)∈[0,𝑇)×𝐂𝑛,(5.39)
where
Ψ𝑇,𝜑(𝑛)𝜑=𝑓(𝑛)∀𝜑(𝑛)∈𝐂𝑛(5.40)
has a solution of the form
𝑉𝑡,𝜑𝑡𝑛𝐹𝑡,𝜑(𝑛).(5.41)
Here, 𝑉(𝑡,(𝜑𝑡)𝑛)=𝑤𝑛(𝑡,0) is a solution to
𝐹𝑡,𝜑(𝑛)𝜕𝑤𝑛𝜕𝑡(𝑡,𝑢)+𝐹𝑡,𝜑𝑡𝑛𝜕𝑤𝑛𝜕𝑢(𝑡,𝑢)(5.42)+𝐆(𝐹)𝑡,𝜑𝑡𝑛𝑤𝑛(𝑡,𝑢)=0(5.43)
for 𝑡∈[0,𝑇] and 𝑢∈[0,𝜖) for some 𝜖>0 and 𝑤𝑛(𝑇,0)=1, and 𝐹∶ℜ+×𝐂𝑛→ℜ is a solution of
𝑟𝐹𝑡,𝜑(𝑛)=𝜕𝐹𝜕𝑡𝑡,𝜑(𝑛)+𝐷𝐹𝑡,𝜑(𝑛)𝑟𝜑(𝑛)(0)𝟏{0}+12𝐷2𝐹𝑡,𝜑(𝑛)𝑁𝜑(𝑛)𝟏{0}𝜑,𝑁(𝑛)𝟏{0}∀𝑡,𝜑(𝑛)∈[0,𝑇)×𝐂𝑛,(5.44)
where
𝐹𝑇,𝜑(𝑛)𝜑=𝑓(𝑛)∀𝜑(𝑛)∈𝐂𝑛,(5.45)
and 𝑓 is a uniformly bounded 𝐶2(𝐂) function satisfying the conditions of Theorem 4.1.

Proof. We assume a solution of the form Ψ(𝑡,𝜑(𝑛))=𝑉(𝑡,(𝜑𝑡)𝑛)𝐹(𝑡,𝜑(𝑛)), then𝑟𝑉𝑡,𝜑𝑡𝑛𝐹𝑡,𝜑(𝑛)=𝜕𝑉𝜕𝑡𝑡,𝜑𝑡𝑛𝐹𝑡,𝜑(𝑛)+𝐆(𝑉𝐹)𝑡,𝜑𝑡𝑛+𝐷(𝑉𝐹)𝑡,𝜑(𝑛)𝑟𝜑(𝑛)(0)𝟏{0}+12𝐷2(𝑉𝐹)𝑡,𝜑(𝑛)𝑁𝜑(𝑛)𝟏{0}𝜑,𝑁(𝑛)𝟏{0}=𝐹𝑡,𝜑(𝑛)𝜕𝑉𝜕𝑡𝑡,𝜑𝑡𝑛+𝑉𝑡,𝜑𝑡𝑛𝜕𝐹𝜕𝑡𝑡,𝜑(𝑛)+𝐹𝑡,𝜑(𝑛)𝐆(𝑉)𝑡,𝜑𝑡𝑛+𝑉𝑡,𝜑𝑡𝑛𝐆(𝐹)𝑡,𝜑(𝑛)+𝑉𝑡,𝜑𝑡𝑛×𝐷(𝐹)𝑡,𝜑(𝑛)𝑟𝜑(𝑛)(0)𝟏{0}+12𝐷2(𝐹)𝑡,𝜑(𝑛)𝑁𝜑(𝑛)𝟏{0}𝜑,𝑁(𝑛)𝟏{0},∀𝑡,𝜑(𝑛)∈[0,𝑇)×𝐂𝑛.(5.46)
If 𝐹(𝑡,𝜑(𝑛)) is the solution to (5.44), then
𝐹𝑡,𝜑(𝑛)𝜕𝑉𝜕𝑡𝑡,𝜑𝑡𝑛+𝐹𝑡,𝜑𝑡𝑛𝐆(𝑉)𝑡,𝜑𝑡𝑛+𝐆(𝐹)𝑡,𝜑𝑡𝑛𝑉𝑡,𝜑𝑡𝑛=0.(5.47)
Define 𝑇𝑢∶𝐂→𝐂 by 𝑇𝑢(𝜑)=𝜑𝑢, that is, 𝑇𝑢 is a shift operator. Now let 𝑉(𝑡,(𝜑𝑡+𝑢)𝑛)=𝑉(𝑡,(𝑇𝑢(𝜑𝑡))𝑛)=𝑤𝑛(𝑡,𝑢) for a fixed 𝜑∈𝐂. Then
𝐆(𝑉)𝑡,𝜑𝑡𝑛=𝜕𝑤𝑛𝜕𝑢𝑡,0+,(5.48)
where the superscript + denotes a right-hand derivative with respect to 𝑢. Then
𝐹𝑡,𝜑(𝑛)𝜕𝑤𝑛𝜕𝑡(𝑡,0)+𝐹𝑡,𝜑𝑡𝑛𝜕𝑤𝑛𝜕𝑢𝑡,0++𝐆(𝐹)𝑡,𝜑𝑡𝑛𝑤𝑛(𝑡,0)=0.(5.49)A slightly more restrictive, but more familiar form is
𝐹𝑡,𝜑(𝑛)𝜕𝑤𝑛𝜕𝑡(𝑡,𝑢)+𝐹𝑡,𝜑𝑡𝑛𝜕𝑤𝑛𝜕𝑢(𝑡,𝑢)+𝐆(𝐹)𝑡,𝜑𝑡𝑛𝑤𝑛(𝑡,𝑢)=0,(5.50)
where 𝑡∈[0,ℎ] and 𝑢∈[0,𝜖) for some 𝜖>0. There is the additional requirement that 𝑤𝑛(𝑇,0)=1 so that (5.44) holds.

Remark 5.3. It can be easily shown that 𝑆𝑡 is 𝛼-Hölder continuous a.s. for 0<𝛼<1/2 provided that 𝑆0 is 𝛼-Hölder continuous for the same 𝛼. Therefore,
||𝐹𝑛||𝑡,⃗𝑥−𝐹(𝑡,𝜑)⟶0(5.51)
for each 𝑡 as 𝑛→∞ where 𝐹(𝑡,𝜑) is a solution to (5.44) and 𝐹𝑛(𝑡,⃗𝑥)=𝐹(𝑡,𝜑(𝑛)) is an approximate solution, since 𝐹 is 𝐶2(𝐂) in its second variable and
𝐹𝑛𝑡,⃗𝑥=𝐹𝑡,𝑒𝑛⃗𝑥=𝐹𝑡,𝜑(𝑛).(5.52)

The proof of the following corollary is identical to that of Theorem 5.2, with the use of Remark 5.3 to obtain Ψ(𝑡,𝜑).

Corollary 5.4. If 𝑆0 is Hölder continuous, then
𝜕𝑟Ψ(𝑡,𝜑)=𝜕𝑡Ψ(𝑡,𝜑)+𝐆(Ψ)𝑡,𝜑𝑡+𝐷Ψ(𝑡,𝜑)𝑟𝜑(0)𝟏{0}+12𝐷2Ψ(𝑡,𝜑)𝑁(𝜑)𝟏{0},𝑁(𝜑)𝟏{0}[,∀(𝑡,𝜑)∈0,𝑇)×𝐂+,(5.53)
where
Ψ(𝑇,𝜑)=𝑓(𝜑)∀𝜑∈𝐂+(5.54)
has a solution of the form 𝑉(𝑡,𝜑𝑡)𝐹(𝑡,𝜑). Here, 𝑉(𝑡,𝜑𝑡)=𝑤(𝑡,0) is a solution to
𝐹(𝑡,𝜑)𝜕𝑤𝜕𝑡(𝑡,𝑢)+𝐹𝑡,𝜑𝑡𝜕𝑤𝜕𝑢(𝑡,𝑢)+𝐆(𝐹)𝑡,𝜑𝑡𝑤(𝑡,𝑢)=0(5.55)
for 𝑡∈[0,𝑇] and 𝑢∈[0,𝜖) for some 𝜖>0. 𝐹(𝑡,𝜑) is the solution to (5.44) where one lets 𝑛→∞. In addition, 𝑤(𝑇,0)=1.

Now we must solve (5.44), which is done in the following theorem. With this solution, the first-order partial differential equation can be solved by traditional means.

Theorem 5.5. Let
𝑟Ψ𝑡,𝜑(𝑛)=𝜕Ψ𝜕𝑡𝑡,𝜑(𝑛)+𝐷Ψ𝑡,𝜑(𝑛)𝑟𝜑(𝑛)(0)𝟏{0}+12𝐷2Ψ𝑡,𝜑(𝑛)𝑁𝜑(𝑛)𝟏{0}𝜑,𝑁(𝑛)𝟏{0}∀𝑡,𝜑(𝑛)∈[0,𝑇)×𝐂𝑛,(5.56)
where
Ψ𝑇,𝜑(𝑛)𝜑=𝑓(𝑛)∀𝜑(𝑛)∈𝐂𝑛,(5.57)
and 𝑓 is a uniformly bounded 𝐶2(𝐂) function satisfying the conditions of Theorem 4.1. Let 𝑓𝑛∶ℜ𝑛+2→ℜ be defined by 𝑓𝑛=𝑓∘𝑒𝑛, then
Ψ𝑛=𝑒𝑡,⃗𝑥−𝑟(𝑇−𝑡)√2𝜋∞−∞𝑓𝑛𝛿exp𝑟𝐵−22𝐵2√(𝑇−𝑡)+𝛿𝐵𝑦𝑒𝑇−𝑡⃗𝑥−𝑦2/2+𝑑𝑦𝑇𝑡𝑔𝑛𝑠,𝜑𝑠(𝑛)𝑒,⃗𝑥(𝑠),Ψ−𝑟(𝑠−𝑡)𝑑𝑠.(5.58)
Here, ⎡⎢⎢⎢⎣𝑘𝐵=0𝑘0⋯𝑘0𝑘1𝑘1⋯𝑘1𝑘⋮⋮⋮⋮𝑛+1𝑘𝑛+1⋯𝑘𝑛+1⎤⎥⎥⎥⎦,𝑣𝑛=𝑛+1𝑖=0𝑘𝑖𝑓𝑖(5.59)
from (5.36).

Proof. Since Ψ𝑛∶[0,𝑇]×ℜ𝑛+2→ℜ, the definition of the Fréchet derivatives and the properties of the set {𝑓𝑖}∞𝑖=0 give
𝑟Ψ𝑛=𝜕𝑡,⃗𝑥Ψ𝜕𝑡𝑛𝑡,⃗𝑥+𝑟𝑛+1𝑖=0𝑥𝑖𝑛+1𝑖=0𝑘𝑖𝜕𝜕𝑥𝑖Ψ𝑛+𝛿𝑡,⃗𝑥22𝑛+1𝑖=0𝑥𝑖2𝑛+1𝑖,𝑗=0𝑘𝑖𝑘𝑗𝜕2𝜕𝑥𝑖𝜕𝑥𝑗Ψ𝑛𝑡,⃗𝑥+𝑔𝑛𝑡,𝜑𝑠(𝑛)∈[,⃗𝑥,Ψ,∀𝑡,⃗𝑥0,𝑇)×ℜ𝑛+2,Ψ𝑇,⃗𝑥(𝑇)=𝑓𝑛.⃗𝑥(𝑇)(5.60)
The ⃗𝑥(𝑇) consists of the first 𝑛+2 coefficients of 𝑆𝑇 in the set of functions {𝑓𝑖}. By the Feynman-Kac theorem (see [15, Theorem 5.7.6]),
Ψ𝑛𝑡,⃗𝑥=𝑒−𝑟(𝑇−𝑡)𝐄𝑓+⃗𝑥(𝑇)∣⃗𝑥(𝑡)=⃗𝑥𝑇𝑡𝑔𝑛𝑠,𝜑𝑠(𝑛)𝑒,⃗𝑥(𝑠),Ψ−𝑟(𝑠−𝑡)𝑑𝑠,(5.61)
where ⃗𝑥(𝑡) is the solution to
𝑑𝑥𝑖(𝑡)=𝑏𝑖𝑡,⃗𝑥(𝑡)𝑑𝑡+𝑛+1𝑗=0𝜎𝑖𝑗𝑑𝑊𝑡,⃗𝑥(𝑡)(𝑗)(𝑡)(5.62)
for 𝑖=0,1,…,𝑛+1. Noting that ⃗𝑥(𝑡)=⃗𝑥,
𝑏𝑖𝑡,⃗𝑥=𝑟𝑘𝑖𝑛+1𝑖=0𝑥𝑖,𝛿2𝑘𝑖𝑘𝑗𝑛+1𝑖=0𝑥𝑖2=𝑛+1𝑘=0𝜎𝑖𝑘𝜎𝑡,⃗𝑥𝑗𝑘𝑡,⃗𝑥(5.63)
with 0≤𝑖,𝑗≤𝑛+1. Hence,
⃗𝑏⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑘𝑡,⃗𝑥=𝑟0𝑛+1𝑖=0𝑥𝑖𝑘1𝑛+1𝑖=0𝑥𝑖⋮𝑘𝑛+1𝑛+1𝑖=0𝑥𝑖⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(5.64)
and so